The University of Montana

Department of Mathematical Sciences

Technical report #26/2008

An Iterative Method for Edge-Preserving MAP Estimation when Data-Noise is Poisson

**John Bardsley**

University of Montana

&

**John Goldes**

University of Montana

**Abstract**

In numerous applications of image processing, e.g. astronomical and medical imaging, data-noise is well-modeled by a Poisson distribution. This motivates the use of the negative-log Poisson likelihood function for data fitting. (The fact that application scientists in both astronomical and medical imaging regularly choose this function for data fitting provides further motivation.) However difficulties arise when the negative-log Poisson likelihood is used. Chief among them are the facts that it is non-quadratic and is defined only for vectors with nonnegative values. The nonnegatively constrained, convex optimization problems that arise when the negative-log Poisson likelihood is used are therefore more challenging than when least squares is the fit-to-data function.

Edge preserving deblurring and denoising has long been a problem of keen interest in the image processing community. While total variation regularization is the gold standard for such problems, its use yields computationally intensive optimization problems. This motivates the desire to develop regularization functions that are edge preserving, but are less difficult to use. We present one such regularization function here. This function is quadratic, and can be viewed as the discretization of a diffusion operator with a diffusion function that is approximately 1 in smooth regions of the true image and is less than 1 (but still positive) at or near an edge.

Combining the negative-log Poisson likelihood function with this quadratic, edge preserving regularization function yields a strictly convex, nonnegatively constrained optimization problem. A large portion of this paper is dedicated to the presentation of and convergence proof for an algorithm designed for this problem. Finally, we apply the algorithm to synthetically generated data in order to test the methodology.

**Keywords:** edge-preserving regularization, inverse problems, nonnegatively constrained optimization, Bayesian statistical methods.

**AMS Subject Classification:** 15A29, 65K10, 65F22.

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